Vol. 15, No. 3, 2021

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The Hodge ring of varieties in positive characteristic

Remy van Dobben de Bruyn

Vol. 15 (2021), No. 3, 729–745

Let k be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers hi,j(X) = dimHj(X,ΩXi) that hold for every smooth proper variety X over k are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of X are given by the span of the hi,0(X) and the h0,j(X) (and their duals hi,n(X) and hn,j(X)). The corresponding statements for compact Kähler manifolds were proven by Kotschick and Schreieder.

algebraic geometry, positive characteristic, Hodge cohomology, de Rham cohomology, Grothendieck ring of varieties, birational invariants
Mathematical Subject Classification 2010
Primary: 14G17
Secondary: 14A10, 14E99, 14F40, 14F99
Received: 28 January 2020
Revised: 28 June 2020
Accepted: 7 September 2020
Published: 20 May 2021
Remy van Dobben de Bruyn
Princeton University
Princeton, NJ
United States
Institute for Advanced Study
Princeton, NJ
United States